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\begin{document}

\title{高等代数一}
\subtitle{26-习题与问答-过渡矩阵-同构-矩阵的秩-基础解系 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
%\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
\date{{\ppr 2022年12月22日} }

\maketitle

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\begin{enumerate}

\item  从一个基到另一个基的过渡矩阵
\item  向量空间的同构
\item  行空间与行秩
%\item  列空间与列秩
\item  齐次线性方程组的基础解系与解空间
%\item  非齐次线性方程组的解集

\end{enumerate}


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{\small 
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
4-习题&8-习题&12-习题&16-习题&21-习题&26-习题 \\ \hline 
{01}&{02}&03&04&05&\underline{06} \\   
{07}&{08}&09&10&11&\underline{12} \\  
{13}&{14}&15&16&17&\underline{18} \\ 
{19}&{20}&21&22&23&\underline{24} \\  
{25}&{26}&27&28&29&\underline{30} \\  
{31}&{32}&33&34&35&\underline{36} \\  
{37}&{38}&39&40&41&\underline{42} \\  
{43}&{44}&45&46&47&\underline{48} \\ 
{49}&{50}&51&52&53&\underline{54} \\  
\end{tabular}
\end{table}
}

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\begin{itemize}

\item  习题1：设 $U$ 与 $W$ 是向量空间 $V$ 的两个子空间，设 $V=U+W$ 且 $U\cap W=\{\theta\}$. 证明对任意 $\alpha\in V$, 存在唯一的 $\beta\in U$ 与唯一的 $\gamma\in W$, 使得 $\alpha = \beta + \gamma$. 

\item  解答思路：
\begin{enumerate}
\item  根据和子空间的定义可得存在性。
\item  从条件 $U\cap W=\{\theta\}$ 得出唯一性。
\end{enumerate}

\end{itemize}

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\begin{itemize}

\item  习题2：设 $V$ 是 $n$ 维实向量空间。设 $\Phi=(\alpha_1, \alpha_2, \cdots, \alpha_n)$ 是 $V$ 的一个基。
设 $\Psi=(\beta_1, \beta_2, \cdots, \beta_n)$ 是 $V$ 中的任意一个向量组。
设 $A$ 是一个 $n$ 阶实数矩阵，使得 $\Psi = \Phi\cdot A$, 即 
{\footnotesize 
\begin{eqnarray*}
\begin{pmatrix} \beta_1 & \beta_2 & \cdots & \beta_n \end{pmatrix} 
=\begin{pmatrix} \alpha_1 & \alpha_2 & \cdots & \alpha_n \end{pmatrix} 
\cdot\begin{pmatrix} 
a_{11} & a_{12} & \cdots & a_{1n} \\ 
a_{21} & a_{22} & \cdots & a_{2n} \\ 
\vdots & \vdots &  &\vdots \\ 
 a_{n1} & a_{n2} & \cdots & a_{nn} \\ 
 \end{pmatrix}. 
\end{eqnarray*}
}
证明向量组 $\Psi$ 的秩等于矩阵 $A$ 的秩。 

\item  解答思路：
\begin{enumerate}
\item  将矩阵 $A$ 化为相抵标准形。
\item  设 $P$ 是可逆矩阵，则向量组 $\Psi\cdot P$ 的秩与 $\Psi$ 的秩相等。 
\end{enumerate}

\end{itemize}

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\begin{itemize}

\item  习题3：设 $f:A\to B$ 与 $g:B\to C$ 是两个映射。设 $h=g\circ f: A\to C$ 是复合映射。证明：
 \begin{enumerate}
\item  如果 $h$ 是单射，那么  $f$ 也是单射。
\item  如果 $h$ 是满射，那么 $g$ 也是满射。
\end{enumerate}

\item  解答思路：根据单射与满射的定义验证。

\end{itemize}

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\begin{itemize}

\item  习题4：设 $f:U\to V$ 与 $g:V\to W$ 是向量空间之间的两个同构。证明复合映射 $h=g\circ f: U\to W$ 也是同构。

\item  解答思路：验证同构的定义里的条件。

\end{itemize}

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\begin{itemize}

\item  习题5：设向量空间 $V=\mathbb{R}^4$ 中的一些向量如下，
{\footnotesize 
\begin{eqnarray*}
%\left\{\begin{array}{l}
\alpha_1=(2,1,-1,1), \,\,
\alpha_2=(0,3,1,0), \,\,
\alpha_3=(5,3,2,1), \,\, 
\alpha_4=(6,6,1,3). 
%\end{array}\right. 
\end{eqnarray*}
}

\vspace{-0.7cm}

\begin{enumerate}
\item  证明向量组 $\Phi=(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ 是 $V$ 的一个基。
\item  求 $V$ 中的向量 $\xi$ 使其关于基 $\Phi$ 的坐标与关于标准基 $\mathcal{E}$ 的坐标是一样的。
\end{enumerate}

\item  解答思路：
\begin{enumerate}
\item  证明向量组 $\Phi$ 线性无关。
\item  设向量  $\xi$ 关于标准基的坐标是 $(x_1,x_2,x_3,x_4)^t$. 使用从基 $\mathcal{E}$ 到基 $\Phi$ 的过渡矩阵求出这个向量关于基 $\Phi$ 的坐标。所求向量为 {\footnotesize $(-k,-k,-k,k)$}, 其中 $k$ 为任意实数。
\end{enumerate}

\end{itemize}

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\begin{itemize}

\item  习题6：求下述子空间的维数：
\begin{enumerate}
\item  $L(x-1, 1-x^2, x^2-x) \subseteq \mathbb{R}[x]$. 
\item  $L(e^x, e^{2x}, e^{3x})\subseteq C[a,b]$. 
\end{enumerate}

\item  解答思路：
\begin{enumerate}
\item  一种方法是建立 $\mathbb{R}[x]_2$ 与 $\mathbb{R}^3$ 的一个同构。维数为2. 
\item  验证这个向量组是线性无关的。维数为3. 
\end{enumerate}

\end{itemize}

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\begin{itemize}

\item  习题7：设 $W$ 是 $V=\mathbb{R}^n$ 的一个真子空间。设 $W$ 中的每个非零向量的分量都不等于零。证明 $W$ 是一维子空间。

\item  解答思路：一种方法是反证法。设 $\dim W\ge 2$, 则有向量 $\alpha,\beta\in W$, 不成比例。想办法构造 $k\alpha+m\beta$, 它既不是零向量，又有分量等于零。

\end{itemize}

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\begin{itemize}

\item  习题8：设 $A$ 与 $B$ 是同阶的矩阵。证明 $R(A+B)\le R(A)+R(B)$. 

\item  解答思路：使用行空间和行秩的概念。

\end{itemize}

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\begin{itemize}

\item  习题9：求齐次线性方程组的系数矩阵的行最简形，并求基础解系，
{\footnotesize 
\begin{eqnarray*}
\left\{\begin{array}{rrrrrr}
x_1& + x_2& -2 x_3& + 2x_4 &=& 0, \\  
3x_1& + 5x_2& + 6x_3& -4 x_4 &=& 0, \\  
4x_1& + 5x_2& -2 x_3& + 3x_4 &=& 0, \\  
3x_1& + 8x_2& + 24x_3& -19 x_4 &=& 0. 
\end{array}\right. 
\end{eqnarray*}
}

\item  答案：系数矩阵的行最简形为 
{\footnotesize 
$%\begin{eqnarray*}
RREF(A) = \begin{pmatrix}  1&0&-8&7 \\  0&1&6&-5 \\  0&0&0&0 \\  0&0&0&0 \\   \end{pmatrix}. 
$%\end{eqnarray*}
}

基础解系为 $ \eta_1 = (8, -6, 1, 0), \,\,  \eta_2 = (-7, 5, 0, 1)$. 

\end{itemize}

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\begin{itemize}

\item  习题10：设矩阵 $A$ 有 $m$ 行，设 $R(A)=r$. 设矩阵 $B$ 由矩阵 $A$ 中的任意 $s$ 行组成。证明 $R(B)\ge r+s-m$. 

\item  解答思路：一个向量组去掉一个向量（或者一个矩阵去掉一行），它的秩最多减去1. 

\end{itemize}

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